Hereditary crossed products

Haefner, Jeremy; Janusz, Gerald

doi:10.1090/S0002-9947-00-02476-4

Hereditary crossed products
HTML articles powered by AMS MathViewer

by Jeremy Haefner and Gerald Janusz PDF

Trans. Amer. Math. Soc. 352 (2000), 3381-3410 Request permission

Abstract:

We characterize when a crossed product order over a maximal order in a central simple algebra by a finite group is hereditary. We need only concentrate on the cases when the group acts as inner automorphisms and when the group acts as outer automorphisms. When the group acts as inner automorphisms, the classical group algebra result holds for crossed products as well; that is, the crossed product is hereditary if and only if the order of the group is a unit in the ring. When the group is acting as outer automorphisms, every crossed product order is hereditary, regardless of whether the order of the group is a unit in the ring.

References

Gene Abrams and Jeremy Haefner, Picard groups and infinite matrix rings, Trans. Amer. Math. Soc. 350 (1998), no. 7, 2737–2752. MR 1422591, DOI 10.1090/S0002-9947-98-01942-4
L. Le Bruyn, M. Van den Bergh, and F. Van Oystaeyen, Graded orders, Birkhäuser Boston, Inc., Boston, MA, 1988. MR 1003605, DOI 10.1007/978-1-4612-3944-4
S. U. Chase, D. K. Harrison, and Alex Rosenberg, Galois theory and Galois cohomology of commutative rings, Mem. Amer. Math. Soc. 52 (1965), 15–33. MR 195922
Charles W. Curtis and Irving Reiner, Methods of representation theory. Vol. I, Pure and Applied Mathematics, John Wiley & Sons, Inc., New York, 1981. With applications to finite groups and orders. MR 632548
M. Postnikov, Lie groups and Lie algebras, “Mir”, Moscow, 1986. Lectures in geometry. Semester V; Translated from the Russian by Vladimir Shokurov. MR 905471
Jeremy Haefner, Graded equivalence theory with applications, J. Algebra 172 (1995), no. 2, 385–424. MR 1322411, DOI 10.1016/S0021-8693(05)80009-2
Jeremy Haefner, Graded Morita theory for infinite groups, J. Algebra 169 (1994), no. 2, 552–586. MR 1297162, DOI 10.1006/jabr.1994.1297
Jeremy Haefner, Reduction techniques for strongly graded rings and finite representation type. I. Bounded functors and graded equivalence, J. Algebra 194 (1997), no. 2, 567–593. MR 1467167, DOI 10.1006/jabr.1997.7031
G. J. Janusz, Tensor products of orders, J. London Math. Soc. (2) 20 (1979), no. 2, 186–192. MR 551444, DOI 10.1112/jlms/s2-20.2.186
S. Jøndrup, The group of automorphisms of certain subalgebras of matrix algebras, J. Algebra 141 (1991), no. 1, 106–114. MR 1118318, DOI 10.1016/0021-8693(91)90206-N
C. Năstăsescu, M. Van den Bergh, and F. Van Oystaeyen, Separable functors applied to graded rings, J. Algebra 123 (1989), no. 2, 397–413. MR 1000494, DOI 10.1016/0021-8693(89)90053-7
Donald S. Passman, Infinite crossed products, Pure and Applied Mathematics, vol. 135, Academic Press, Inc., Boston, MA, 1989. MR 979094
M. D. Rafael, Separable functors revisited, Comm. Algebra 18 (1990), no. 5, 1445–1459. MR 1059740, DOI 10.1080/00927879008823975
Neal H. McCoy, The theory of rings, Chelsea Publishing Co., Bronx, N.Y., 1973. Reprint of the 1964 edition. MR 0393090